Let $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$

Is it true or false that a combination of two (or more, in general) of the roots can give us another root?

In mathematical terms, does there exist indices $i_1,i_2,...,i_s, j,$ such that $z_j = \sum_{k=1}^s z_{i_k}?$

It seems to be that this is not possible, but I also don't have proof of that.

Thank you!

Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$

Is it true or false that a combination of two (or more, in general) of the roots can give us another root of the same order?

In mathematical terms, does there exist indices $i_1,i_2,...,i_s, j,$ such that $z_j = \sum_{k=1}^s z_{i_k}?$

It seems to be that this is not possible, but I also don't have proof of that.

Thank you!